Physica A 390 (2011) 2855–2862

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Physica A

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Neural signal transduction aided by noise in multisynaptic excitatoryand inhibitory pathways with saturation

Fabing Duan a,∗, François Chapeau-Blondeau b, Derek Abbott ca College of Automation Engineering, Qingdao University, Qingdao 266071, PR Chinab Laboratoire d’Ingénierie des Systèmes Automatisés (LISA), Université d’Angers, 62 avenue Notre Dame du Lac, 49000 Angers, Francec Centre for Biomedical Engineering (CBME) and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia

a r t i c l e i n f o

Article history:Received 22 November 2010Received in revised form 23 January 2011Available online 12 April 2011

Keywords:Saturating dynamicsInhibitory synapseCorrelation coefficientStochastic resonance

a b s t r a c t

We study the stochastic resonance phenomenon in saturating dynamical models of neuralsignal transduction, at the synaptic stage, wherein the noise in multipathways enhancesthe processing of neuronal information integrated by excitatory and inhibitory synapticcurrents. For an excitatory synaptic pathway, the additive intervention of an inhibitorypathway reduces the stochastic resonance effect. However, as the number of synapticpathways increases, the signal transduction is greatly improved for parallel multipathwaysthat feature both excitation and inhibition. The obtained results lead us to the realizationthat the collective property of inhibitory synapses assists neural signal transmission, anda parallel array of neurons can enhance their responses to multiple synaptic currents byadjusting the contributions of excitatory and inhibitory currents.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The stochastic resonance phenomenon originally describes the match between the noise induced characteristic timescale of a system response and that of an input periodic subthreshold signal [1–4]. Gradually, different forms of stochasticresonance were shown to be feasible, with various types of signals, nonlinear systems, and measures of performancereceiving improvement from noise [2–11]. For instance, suprathreshold stochastic resonance (SSR) occurring in anuncoupled parallel array breaks the restriction of a subthreshold input [7]. Subsequently, the input–output gain is enhancedby noise in array of static [11–16] or dynamical [17,18] nonlinear elements and has been demonstrated to exceed unity,which cannot be obtained by linear systems. Up to the present, the terminology stochastic resonance is used very frequently inthe much wider sense of being the occurrence of any kind of noise-enhanced or noisy constructive phenomena in nonlinearsystems [6]. Thus, stochastic resonance should be regarded as a complex phenomenon, and its key feature is the constructiverole of noise. In this broad sense, many noise-enhanced phenomena such as the noise-induced linearization [19,20] anddithering [21,22] can be also contained within the conceptual framework of stochastic resonance [1–4,6].

Usually, the appearance of stochastic resonance is closely tied to nonlinear systems working in a noisy environment[1–3,6]. Therefore, due to nonlinearity within neurodynamics and the prevalently noisy environment of neural systems, theobservations of stochastic resonance in such single neuronmodels as the FitzHugh–Nagumomodel [3,23–27], integrate-firemodel [28–32], threshold model [33,34,26,35], Hodgkin–Huxley model [36], neural network models [26,37–48] and realneurons [49–56] are progressively and continuously reported. So far, the main scenario of stochastic resonance occurring in

∗ Corresponding author.E-mail addresses: fabing.duan@gmail.com, fabing1974@yahoo.com.cn (F. Duan).

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.03.039

2856 F. Duan et al. / Physica A 390 (2011) 2855–2862

neural systems is the response of a neuron to a threshold or potential-barrier nonlinearity [2,3,23–25,28]. However, anotherform of stochastic resonance is recently reported exploiting a saturating nonlinearity [12,15,57], wherein no threshold orpotential barrier is bestowed. In a recent paper [57], we demonstrated that essentially saturating dynamics, which is presentin synaptic transduction, can give rise to stochastic resonance or an improvement via noise at this stage of neural signaltransmission. Compared to threshold or potential-barrier dynamics, the improvement by noise via saturating dynamicstakes place in more demanding conditions, wherein the inherent fluctuations of spontaneous firing rate or number ofreleased neurotransmitter vesicles realize a non-negative noise representing these non-negative quantities at the synapticstage [57]. This non-negative character of noise is attached to the neural interpretations of several stages of neural signaltransmission [12,15,57,58].

In our previous study of stochastic resonance [57], we only considered a single synaptic pathway modeling the excitablesynapse governed by saturating dynamics. However, realistic models of neurodynamics must ultimately encompassmultiple interacting modules, and the neural information processing is the integration of excitatory and inhibitory synapticcontributions [59–68]. In the present paper, we first consider two synaptic pathways, i.e. one being excitatory and the otherbeing inhibitory. The sum of both excitatory and inhibitory synaptic currents is then taken as the relevant output of thesetwo synaptic pathways. Subsequently, the uncoupled parallel array of multiple synaptic pathways composed of a numberof excitatory synapses and inhibitory synapses is considered. We numerically analyze the constructive role of noise fromthe inhibitory synaptic pathway for neuronal signal transduction, and also investigate the effect of the efficacy parametersof multisynaptic pathway on stochastic resonance or improvement of signal transmission by noise. The obtained resultswill lead to research attention on more elaborate dynamics for information improvement by noise in the nervous systemsoperated with saturating nonlinearity.

2. Model and measure

At the synaptic stage, neural signal transduction relies on neurotransmitter release and gating of ion channels. In the caseof strong sustained presynaptic activity, both neurotransmitter pools and populations of ion channels can all be recruited inthe process, therefore limiting any further increase of the electric current induced across the membrane of the postsynapticneuron. This process can be described by the saturating dynamics [57–61]

dIj(t)dt

= −Ij(t)τ

+ [Isat,j − Ij(t)]wjej(t), (1)

where Ij(t) denotes the synaptic current of an excitatory or inhibitory synapse j, τ represents the relaxation time ofsaturating dynamics, and the saturation current Isat,j is positive for an excitatory synapse j and negative for an inhibitorysynapse j [60–63]. Here, the positive parameter wj measures the efficacy of synapse j in converting the non-negativepresynaptic activity ej(t) into a postsynaptic current Ij(t) [58].

As we have interpreted the saturating dynamics of Eq. (1) in our previous study [57], the input signal ej(t) in Eq. (1)is non-negative, and represents the presynaptic activity at the input of the transmission pathway. Here, we assume thatthese synapses are attached to the same presynaptic cell, and the input signal ej(t) = s(t) + ξj(t) into the synapse j istaken as the superposition of a deterministic aperiodic waveform s(t) and noise ξj(t). The ξj(t) are modeled as independentnon-negative white noise sources with the same probability density function fξ (u) for which various standard forms can beconsidered [57]. Here, we mainly consider the gamma probability density of order a ≥ 1, as

fξ (u) =

1

baΓ (a)ua−1 exp

−

ub

, u ≥ 0,

0, u < 0,(2)

whereΓ (a) =

∞

0 xa−1 exp(−x)dx, themeanvalue of ξj(t)being ab, its variance ab2, and its rms amplitude ξrms = b√a2 + a.

It is noted that the non-negative uniform noise and gamma noise were considered for improving the synaptic signaltransduction, and the similar stochastic resonant behaviors for both kinds of noise have been observed in a single excitatorysynapse [57].

For multiple synapses, the total synaptic current collecting several synaptic currents Ij(t) as

I(t) =

−j

Ij(t), (3)

is taken as the relevant output of this multisynaptic pathway. The total synaptic current I(t) forms the somatic current,which directly drives the dynamics of the membrane potential of the postsynaptic neuron. When this potential reaches thefiring threshold, a spike is emitted by the neuron and its potential is reset to its resting value. This stage constitutes thewell-known threshold dynamics of the neuron, which has been shown to lend itself to various forms of stochastic resonance [2–4,6,18,23–25,28,29,31,33,34,26,27]. Yet here we are not considering this threshold nonlinear dynamics of the neuron. Instead,we investigate the nonlinear saturating dynamics of Eq. (1), which takes place in synaptic transduction. Then, in orderto investigate the possibilities of noise-aided signal transmission in such multisynaptic pathways, a useful input–outputmeasure of similarity, frequently used in stochastic resonance studies, is the correlation coefficient [23,26,27,52,61,62] of

F. Duan et al. / Physica A 390 (2011) 2855–2862 2857

Fig. 1. An example of the input signal s(t) from Eq. (5) with Ts = 103τ , A1 = 5Isat,j, A2 = 3Isat,j and A3 = 2Isat,j . Here, the relaxation time of saturatingdynamic τ = 0.1s and the saturation current Isat,j = 1 in Eq. (1).

the input s(t) with the output I(t)

ρsI =[s(t) − s(t)][I(t) − I(t)]

[s(t) − s(t)]2[I(t) − I(t)]21/2 , (4)

where the overbar denotes a temporal average.

3. Simulation results

Based on the neural interpretation of Eq. (1), the input ej(t) into the synapse j is non-negative, and taken as thesuperposition ej(t) = s(t) + ξj(t) of a deterministic aperiodic waveform s(t) and noise ξj(t). Both components s(t) andξj(t) are separately formed with the same neural substrate as ej(t), and share the same nature of non-negative quantitiesthat describe presynaptic activity, be it a spontaneous random activity for ξj(t) or a coherent activity for s(t) [57]. The non-negative noise ξj(t) is given in Eq. (2), and the deterministic aperiodic waveform s(t) is here defined over t ∈ [0, Ts] as

s(t) = A1 sin(π t/Ts) + A2 sin(3π t/Ts) + A3 sin(7π t/Ts), (5)with s(t) being zero outside [0, Ts]. An example of the waveform s(t) of Eq. (5) is depicted in Fig. 1. The purpose of Eq. (5) isto have a non-negative signal s(t), which carries some distinctive features in the upper part of the waveform that can sufferfrom saturation in transmission by Eq. (1).

When the noise ξj(t) is absent, the solution to Eq. (1) with initial condition Ij(t0) reads

Ij(t) =

Ij(t0) + Isat,j

∫ t

t0wj s(t ′′) exp

t ′′ − t0

τ+

∫ t ′′

t0wj s(t ′)dt ′

dt ′′

× exp[−

t − t0τ

−

∫ t

t0wj s(t ′)dt ′

], (6)

for t ≥ t0. Thus, in the presence of input signal s(t) only, the correlation coefficient ρsI between the somatic currentI(t) =

∑j Ij(t) and the input signal s(t) can then be computed directly according to Eqs. (4) and (6). When the noise

ξj(t) is present at the input, the somatic current I(t) will be simulated numerically [57,69], and an ensemble-averagingof the correlation coefficient ρsI will be performed over independent realizations of the noise to yield ⟨ρsI⟩, following thecommon practice in stochastic resonance studies [23,27]. In numerical simulations of Eq. (1), the Euler–Maruyama methodis used [69], with a sampling time step 1t much less than the time constant τ and the signal duration Ts, and which is fixedat 1t = 0.1τ throughout.

3.1. Two synaptic pathways of an excitatory and an inhibitory

For a single excitatory synapse, the constructive influence of the rms noise amplitude of ξ1(t) on the synaptic currentI(t) = I1(t) has been demonstrated with the appearance of the stochastic resonance effect [57], which is also numericallyshown in Fig. 2(a) denoted by asterisks (∗). Here, the excitatory synapse j = 1 is always subjected to the noisy inpute1(t) = s(t) + ξ1(t), and the efficacy parameter of the synapse w1 = 100. The input signal s(t) is with Ts = 103τ ,A1 = 5Isat,1, A2 = 3Isat,1 and A3 = 2Isat,1. It is clearly visible in Fig. 2(a) that, as the rms noise amplitude ξrms increases, theinput–output ensemble-averaging of correlation coefficient ⟨ρsI⟩ presents a stochastic resonant behavior. In other words,

2858 F. Duan et al. / Physica A 390 (2011) 2855–2862

a b

Fig. 2. Ensemble-averaged correlation coefficient ⟨ρsI ⟩ between the input stimulus s(t) and the somatic current I(t), as a function of the noise rmsamplitude ξrms , for different values of the inhibitory saturation current Isat,2/Isat,1 indicated by the legend. The excitatory presynaptic activity is e1(t) =

s(t) + ξ1(t) and the efficacy parameter of the synapse w1 = 100. The inhibitory presynaptic activity is (a) e2(t) = ξ2(t), and (b) e2(t) = s(t) + ξ2(t). Here,the input signal s(t) from Eq. (5) is with Ts = 103τ , A1 = 5Isat,1, A2 = 3Isat,1 and A3 = 2Isat,1 . The efficacy parameter of the synapse w2 = 50. Here, ξ1(t)and ξ2(t) are independent gamma noise sources of order a = 2 with the same noise rms amplitude ξrms . Each point was averaged over 1000 trials. Forsimplicity, we represent the origin tick 10−3 of the logarithmic x-axis as zero, and ξrms actually increases from zero.

an optimal amount of noise corresponds to the maximum value of ⟨ρsI⟩, while too little or too much noise reduces thecorrelation coefficient ⟨ρsI⟩. For the efficacy parameter of the synapse w1 = 100, the system of Eq. (1) leads to theoperation in the saturation region. It is seen in Fig. 2(a) that, at the level of noise ξrms = 0, the initial correlation coefficient⟨ρsI⟩ = 0.6301 calculated by Eq. (6) is rather lower than unity. This indicates that, due to the saturation dynamics ofEq. (1), the distortion undergone by the input s(t) results in a corresponding distorted waveform of I(t) as compared withs(t). However, as the noise level ξrms is raised above zero, Fig. 2(a) shows that the addition of noise is able to overcome thisdistortion imposed on I(t), and a suitable amount of added noise can pull the output I(t) away from the strong saturationregime, eliciting the possibility of an improvement of ⟨ρsI⟩ by noise in one excitatory synapse [57].

Next, we connect an inhibitory synapse j = 2 to the excitatory synapse j = 1, and the somatic current I(t) is the sum ofthe synapse currents of I1(t) and I2(t) of two synapses j = 1, 2. Since a certain degree of heterogeneity both in their internalparameters and in their connectivity pattern ismore popular in real populations [61], we take different values of the efficacyparameter. Here, the inhibitory synapse j = 2 is with the efficacy of synapse w2 = 50. We observe that:

(i) When the random activity of an inhibitory synapse is e2(t) = ξ2(t), Fig. 2(a) shows the correlation coefficient ⟨ρsI⟩

between the input s(t) contained in the excitatory activity e1(t) and the output I(t) as a function of the rms noise amplitudeξrms, for different values of the saturation current Isat,2/Isat,1 = −1/7, − 3/7, − 1 and −2. Here, ξ1(t) and ξ2(t) areindependent, but with the same rms amplitude ξrms. Since the presynaptic input e2(t) of the inhibitory synapse j = 2 doesnot contain the signal s(t), ⟨ρsI⟩ still starts from the same value of 0.6301 at ξrms = 0 by the excitatory dynamics of Eq. (6)driven by e1(t) = s(t). It is seen that the addition of the random activity of noise e2(t) = ξ2(t) degrades the emergenceof stochastic resonance in the single excitatory synapse j = 1. When the saturation current Isat,2/Isat,1 decreases to −2, thestochastic resonance effect disappears: a monotonic decay of ⟨ρsI⟩ toward zero is observed when the rms noise amplitudeξrms increases.

(ii) When the presynaptic activity of the inhibitory synapse j = 2 is e2(t) = s(t) + ξ2(t), the correlation coefficient ⟨ρsI⟩

between the input signal s(t) and the somatic current I(t) is illustrated in Fig. 2(b) as a function of the rms noise amplitudeξrms. It is observed in Fig. 2(b) that, as the rms noise amplitude ξrms takes the level of zero, the initial ⟨ρsI⟩ takes differentinitial values. The reason is that the inhibitory presynaptic activity e2(t) also receives the input signal s(t), and initial ⟨ρsI⟩

should be calculated by summing the excitatory and inhibitory dynamics of Eq. (6) driven by e1(t) = e2(t) = s(t). For thepositive saturation current Isat,1, the output I1(t) is predicatively positive, and partly similar to the input s(t). Contrarily,the output I2(t) is negative and partly similar to −s(t), as the saturation current Isat,2 is always negative for the inhibitorysynapse. Naturally, at the rms noise amplitude ξrms = 0, the somatic current I(t) = I1(t) + I2(t) is usually less similar tothe input s(t), and the initial ⟨ρsI⟩ between I(t) and s(t), as shown in Fig. 2(b), is less than the value of 0.6301 illustratedin Fig. 2(a) that depicts the similarity of I1(t) to s(t). It is also interesting to note that, as the inhibitory saturation currentIsat,2/Isat,1 decreases from−1/7 to−2, ⟨ρsI⟩ exhibits two opposite kinds of stochastic resonance phenomena versus the rmsnoise amplitude ξrms. Here, the smaller the negative value of ⟨ρsI⟩ is, the closer the relationship between I(t) and−s(t). Theseresonant curves of ⟨ρsI⟩ plotted in Fig. 2(b) demonstrate the constructive roles of ξ1(t) and ξ2(t) for the efficacy of signaltransmission through the dynamical saturating synapses in a complex way. Furthermore, as the rms noise amplitude ξrmsincreases to a very large levels, e.g. ξrms/Isat,1 = 102, as shown in Fig. 2(b), the correlation coefficients ⟨ρsI⟩ that correspondto a different saturation current Isat,2 congregate around the region being close to zero. This means that too much noisedisturbs the somatic current I(t), reducing the effect on the input signal s(t) or −s(t).

F. Duan et al. / Physica A 390 (2011) 2855–2862 2859

Fig. 3. Ensemble-averaged correlation coefficient ⟨ρsI ⟩ between input s(t) and output I(t) =∑N

j=1 Ij(t), as a function of the noise rms amplitude ξrms , fordifferent numbers of excitatory synapses N = 1, 2, 10, 500 and 1000 indicated by the legend. Each excitatory synapse j is with the presynaptic activityej(t) = s(t) + ξj(t) and the efficacy parameter of the synapse wj = 100. No inhibitory synapse is connected. For simplicity, we represent the origin tick10−3 of the logarithmic x-axis as zero, and ξrms actually increases from zero. Other parameters are the same as in Fig. 2.

3.2. Parallel arrays of excitatory synapses and inhibitory synapses

Next, we investigate the improvement of signal transduction through a population of synapses via the constructive roleof internal noise in each synapse. In particular, the study of signal transmission in parallel arrays of nonlinear systems hasreceived considerable attention, and in such arrays, an important specific form of stochastic resonance, i.e. suprathresholdstochastic resonance, has been reported [7]. Moreover, the essential role of neuronal noise is subsequently investigated forinformation transmission in the context of suprathreshold stochastic resonance [37,38]. Here, we also consider a populationof synapses on the same postsynaptic neuron that realizes a real-world example of the basic structure of suprathresholdstochastic resonance, aiming to observe a more remarkable improvement in a parallel bundle of synapses [26,27,39,60–63].

We first consider a parallel array ofmultiple synapses consisting of excitatory synapses for j = 1, 2, . . . ,N with the samesaturation current Isat,j, and each is driven by ej(t) = s(t) + ξj(t). No inhibitory synapses are connected. Then, the collectivecurrent of synaptic pathways is I(t) =

∑Nj=1 Ij(t). Fig. 3 shows the correlation coefficient ⟨ρsI⟩ as a function of the rms noise

amplitude ξrms, for different numbers of excitatory synapses N = 1, 2, 10, 500 and 1000. As the rms noise amplitude ξrmsincreases from zero, the feature of stochastic resonance is obviously demonstrated by the bell-type curve of ⟨ρsI⟩ versus ξrms.It is seen in Fig. 3 that, as the number of excitatory synapses N increases, the ensemble collective property of noise ξj(t) ineach excitatory synapses j = 1, 2, . . . ,N can further enhance the correlation coefficient ⟨ρsI⟩. For instance, the maximumof ⟨ρsI⟩ can be improved to 0.8001 at ξrms/Isat,j = 1.12 for a parallel array of N = 1000 excitatory synaptic pathways. Inview of the input signal s(t) being with A1 = 5Isat,j, A2 = 3Isat,j and A3 = 2Isat,j, this enhancement effect of ⟨ρsI⟩ by boththe noise ξj(t) and the parallel array size N can be regarded as a specific form of suprathreshold stochastic resonance [7] ina broad sense. The obtained results of Fig. 3 also indicate the strategy of assembling excitatory synaptic pathways into anuncoupled parallel array to improve signal transduction at the synaptic stage is possible and advantageous.

We further explore the signal transduction in a parallel array of multiple synapses composed of both excitatory synapsesj = 1, 2, . . . ,N , with the same saturation current Isat,j, and inhibitory synapses k = 1, 2, . . . ,M , with the same saturationcurrent Isat,k. Then, the somatic current becomes I(t) =

∑Nj=1 Ij(t) +

∑Mk=1 Ik(t). Correspondingly, ξj(t) and ξk(t) are also

mutually independent, but with a same rms noise amplitude ξrms. Here, the excitatory synapses are always subjected to thenoisy input stimuli ej(t) = s(t) + ξj(t), while two kinds of the inhibitory synaptic input ek(t) are considered:

(i) First, each inhibitory synapse k is driven by the random activity of ek(t) = ξk(t) for k = 1, 2, . . . ,M . Thecorresponding correlation coefficient ⟨ρsI⟩ versus the noise amplitude ξrms is shown in Fig. 4. In Fig. 4(a), the saturationcurrent Isat,k/Isat,j = −5/7 is fixed, while the numbers of synapses N = M increase from unity to 1000. It is seen inFig. 4(a) that the array stochastic resonance effect is clearly visible: The larger the number of the parallel arrays of synapticpathways is, the more prominent the correlation coefficient ⟨ρsI⟩ is enhanced by the noise sources ξj(t) and ξk(t). It is seenthat when the numbers of the multiple synaptic pathways increase as N = M ≥ 500, the curves of ⟨ρsI⟩ converge. For themultiple synaptic pathways with large synaptic numbers N = M = 1000 (down triangles), the maximum of ⟨ρsI⟩ can beenhanced to 0.886 at ξrms/Isat,j = 2.5. Compared with Fig. 3, obtained by only the multiple excitatory synaptic pathways,this improvement of ⟨ρsI⟩ of 0.886 is quite evident. Furthermore, we plot the correlation coefficient ⟨ρsI⟩ versus the rmsnoise amplitude ξrms in Fig. 4(b) for different inhibitory saturation currents Isat,k/Isat,j and the same numbers of synapsesN = M = 1000. It is shown in Fig. 4(b) that, for the inhibitory saturation current Isat,k/Isat,j = −1 (squares), the maximumof ⟨ρsI⟩ can be further improved to 0.9241 at ξrms/Isat,1 = 9.7, whereby the collective effect of synaptic noise sources ξj(t)and ξk(t) on the correlation coefficient ⟨ρsI⟩ is optimized. It is observed in Fig. 4(b) that other values of Isat,2/Isat,1 cannotacquire this optimal value of the correlation coefficient ⟨ρsI⟩. Finally, the inhibitory saturating current Isat,k/Isat,j = −1 and

2860 F. Duan et al. / Physica A 390 (2011) 2855–2862

a

c

b

Fig. 4. Ensemble-averaged correlation coefficient ⟨ρsI ⟩ between input s(t) and the somatic current I(t) =∑N

j=1 Ij(t)+∑M

k=1 Ik(t), as a function of the noiserms amplitude ξrms . Here, the inhibitory pathway is combined in parallel with the excitatory pathway, resulting in parallel arrays of multiple synapses.Note that each inhibitory synapse k is purely driven by the random activity of ek(t) = ξk(t) for k = 1, 2, . . . ,M , while each excitatory synapse j iswith the presynaptic activity ej(t) = s(t) + ξj(t). The efficacy parameters of the synapse are wj = 100 and wk = 50, respectively. (a) The numbersof synapses are N = M = 1, 2, 10, 100, 500 and 1000, while the saturation current Isat,k/Isat,j = −5/7 is fixed; (b) The inhibitory saturation currentsIsat,k/Isat,j = −1/7, −3/7, −5/7, −1, −1.1 and −2, but the numbers of synapses N = M = 1000 are fixed; (c) The inhibitory saturation currentIsat,k/Isat,j = −1 and the number of excitatory synapses N = 1000 are fixed, while the number of inhibitory synapsesM varies. For simplicity, we representthe origin tick 10−3 of the logarithmic x-axis as zero, and ξrms actually increases from zero. Other parameters are the same as in Fig. 2.

the number of excitatory synapses N = 1000 are fixed, while the parallel inhibitory synapses vary their connected numberM . The obtained results of ⟨ρsI⟩ versus ξrms are then shown in Fig. 4(c), which indicates the number of M = 1000 (squares)corresponds to the best behavior of ⟨ρsI⟩ as ξrms increases.

In this case of the parallel inhibitory synapses k = 1, 2, . . . ,M driven by pure random activities ek(t) = ξk(t), thesynaptic currents Ik(t) are all negative. When these negative currents Ik(t) are added to the outputs Ij(t) of excitatorysynapses j = 1, 2, . . . ,N , the summed result of somatic current I(t) manifests more similar to the input s(t). This collectiveproperty of ξk(t) in inhibitory synapses seems to counteract the influence of ξj(t) in excitatory synapses on the output I(t),leading to a great improvement of ⟨ρsI⟩ shown in Fig. 4.

(ii) Next, we assume the inhibitory presynaptic activity as ek(t) = s(t) + ξk(t) for k = 1, 2, . . . ,M . Two representativeexamples of the corresponding correlation coefficient ⟨ρsI⟩ are illustrated in Fig. 5(a) and (b). In Fig. 5(a), the inhibitorysaturation currents Isat,k/Isat,j = −1/7, and the behaviors of ⟨ρsI⟩ versus the noise amplitude ξrms are plotted for differentsynaptic numbers N = M = 1, 2, 10, 100 and 1000. Since |Isat,k| is relatively smaller than Isat,j, the initial correlationcoefficient ⟨ρsI⟩ is still positive according to Eq. (6). As the rms noise amplitude ξrms increases, ⟨ρsI⟩ is first raised to themaximum value, and then degrades gradually, which are the typical convex curves of stochastic resonance. Conversely,for the inhibitory saturating currents Isat,k/Isat,j = −1, the initial correlation coefficient ⟨ρsI⟩ is negative. Note that ⟨ρsI⟩ isimproved towards −1 in absolute value as the rms noise amplitude ξrms increases. As we explained, the negative value of⟨ρsI⟩ in Fig. 2(b) denotes that the somatic current of I(t) is similar to the input −s(t). The constructive role of noise andthe collective effect of synaptic numbers N and M on ⟨ρsI⟩ can be also regarded as a kind of stochastic resonance effect.However, in order to obtain the positive values of ⟨ρsI⟩, the case of entering only random activity ek(t) = ξk(t) into theinhibitory synapses, as shown in Fig. 3, is preferable to this case of the inhibitory synapses with the presynaptic activitytaking ek(t) = s(t) + ξk(t) of Fig. 5.

F. Duan et al. / Physica A 390 (2011) 2855–2862 2861

a b

Fig. 5. Ensemble-averaged correlation coefficient ⟨ρsI ⟩ between input s(t) and output I(t), as a function of the noise rms amplitude ξrms . Here, eachinhibitory synapse k is driven by ek(t) = s(t)+ ξk(t) for k = 1, 2, . . . ,M , and each excitatory synapse j is with the presynaptic activity ej(t) = s(t)+ ξj(t).The efficacy parameter of the synapse wj = 100 and wk = 50, respectively. The numbers of synapses N = M = 1, 2, 10, 100, 500 and 1000, while thesaturation current (a) Isat,k/Isat,j = −1/7 and (b) Isat,k/Isat,j = −1. For simplicity, we represent the origin tick 10−3 of the logarithmic x-axis as zero, andξrms actually increases from zero. Other parameters are the same as in Fig. 2.

4. Conclusion

In this paper, we studied the signal transmission in parallel arrays of synapses composed of both the excitatory synapticpathway and the inhibitory synaptic pathway at the synaptic stage. The corresponding synaptic currents from the excitatorysynaptic pathways and the inhibitory synaptic pathways are summed as the somatic current of multiple synaptic pathways.For a single excitatory synapse, the combination of an inhibitory synapse does not enhance the stochastic resonanceeffect. When large numbers of excitatory synapses are assembled as a parallel array, as compared with a single excitatorysynapse, the improvement of the correlation coefficient by noise can be slightly increased. Interestingly, the enhancementof correlation coefficient by noise can be greatly improved by introducing inhibitory pathways into the parallel arraysof multiple synaptic pathways, and the independent noise sources in multiple synaptic pathways play an integratedconstructive role on signal transduction by increasing the synaptic numbers as well as the rms noise amplitude. We arguethat the negative output of inhibitory synapses purely driven by random noise can mostly counteract the influence ofsynaptic noise in excitatory pathways on the somatic current, resulting in the prominent improvement of the input–outputcorrelation coefficient illustrated in Fig. 4. Since processing of neural information usually involves integration of excitatoryand inhibitory synaptic inputs [64,60,61], the observation of stochastic resonance in the noisy multiple synaptic pathwaysconsidered in this paper points to a useful mechanism accessible to neural signals at the synaptic stage. The possibility ofstochastic resonance for neural signal transduction motivates the need for further experiments on live neurons.

Acknowledgments

This work is sponsored by the Taishan Scholar CPSP and the Natural Science Foundation of Shandong Province, China(No. ZR2010FM006). Funding from the Australian Research Council (ARC) is gratefully acknowledged.

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